Minkowski inequality

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In mathematical analysis, the Minkowski inequality establishes that the L^p spaces are normed vector spaces. Let S be a measure space, let 1 ≤ p ≤ ∞ and let f and g be elements of L^p(S). Then f + g is in L^p(S), and we have

with equality for 1<p<∞ only if f and g are linearly dependent.

The Minkowski inequality is the triangle inequality in L^p(S).

Like Hölder's inequality, the Minkowski inequality can be specialized to sequences and vectors by using the counting measure:

for all real (or complex) numbers x₁, ..., x_n, y₁, ..., y_n and where n is the cardinality of S.

[edit] Proof

Choose q so that 1/p + 1/q =1. Then

(because |f+g| ≤ |f|+|g|)

(by Hölder's inequality).

Since (p-1)q = p and 1/q = 1-1/p, we obtain Minkowski's inequality by multiplying both sides by ∫(|f+g|^pdμ)^(1/p)−1.

[edit] References

• G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities , Cambridge Univ. Press (1934) ISBN 0-521-35880-9
• H. Minkowski, Geometrie der Zahlen , Chelsea, reprint (1953)
• M.I. Voitsekhovskii, "Minkowski inequality" SpringerLink Encyclopaedia of Mathematics (2001)